# Symmetry operators for 2nd quantized Hamiltonian

Symmetry operators in for second quantized operators only act on second quantized operators and do not act on numbers (i.e. the ‘first quantized Hamiltonian’, $\mathcal{H}$, or Hamiltonian density), UNLESS the symmetry operator is the time reversal operator, $\mathcal{T}$.

The symmetry operation, $\mathcal{O}$, is defined in terms of second quantized field operators: $\mathcal{O}\psi\mathcal{O}^{-1} = U_{\mathcal{O}} \psi$, where $U_{\mathcal{O}}$ is a unitary matrix.

If a second quantized Hamiltonian, $H=\Psi^\dag \mathcal{H}\Psi$, is invariant under the symmetry operator, $\mathcal{O}$, then we have $[H,\mathcal{O}]=0$. By using the way the symmetry operation acts on the field operators, we can define an effective action of the symmetry operator on the Hamiltonian density, $\mathcal{H}$.

According to the above comment, it is trivial to write something like $\mathcal{O}\mathcal{H}\mathcal{O}^{-1}$ since we always have $\mathcal{O}\mathcal{H}\mathcal{O}^{-1}=\mathcal{H}$, unless when $\mathcal{O}$ is the time reversal operator $\mathcal{T}$, under which circumstance $\mathcal{T}\mathcal{H}\mathcal{T}^{-1}=\mathcal{H}^*$.